Hi Michelle! When you’re logged into the NYT on the web, at the top of the story just above the byline of the author, there are icons — including for commenting on the article or book marking it. There will also be a button with the icon for a gift wrap box and the text “share full article”. Click on that, and it will give you an option to copy the link. You’re allowed to share up to 10 articles per month!
On iOS, the little gift icon is in the bottom right corner of any story that you’re looking at. I’m not sure where it is on android, my apologies, I don’t currently have an android device …